Derivative of \( \displaystyle 5 x \left(25 x^{2} - 45 x + 27\right) \)

Problem 2.1419 · hard

1. \[ \frac{d}{d x} 5 x \left(25 x^{2} - 45 x + 27\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(125 x^{3} - 225 x^{2} + 135 x\right) \]
   algebraDistribute 5*x into the parentheses.✓ Proved
3. \[ = \frac{d}{d x} 135 x - \frac{d}{d x} 225 x^{2} + \frac{d}{d x} 125 x^{3} \]
   sumApply the sum rule for differentiation.✓ Proved
4. \[ = 135 \frac{d}{d x} x - 225 \frac{d}{d x} x^{2} + 125 \frac{d}{d x} x^{3} \]
   constant-multiplePull out the constant coefficients.✓ Proved
5. \[ = 375 x^{2} - 450 x + 135 \]
   power simplifyApply the power rule to each term. Multiply the constants to get the final result.✓ Proved